Nonlinear diffusion filtering on extended neighborhood

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1 Applied Numerical Mathematics 5 005) Nonlinear diffusion filtering on extended neighborhood Danny Barash Genome Diversity Center, Institute of Evolution, University of Haifa, Mount Carmel, Haifa 31905, Israel Available online 17 September 004 Abstract A generalization to the nonlinear diffusion of digital images is proposed, taking into account an extended neighborhood as a prolongation to information derived from local, nearest-neighboring pixels. Traditionally, each iteration in nonlinear diffusion filtering is performed on a 3 3 window. While several iterations can propagate this local information to pixels which are further away from the center of the window, it clearly presents a limitation in some applications as compared to non-pde based filtering approaches in which the window is larger than 3 3. Here, an extension is presented to describe the nonlinear diffusion equation for larger windows, thus overcoming this limitation. 004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Nonlinear diffusion filtering; Adaptive smoothing; Digital TV filtering; Extended neighborhood 1. Introduction The nonlinear diffusion equation as a way to filter images has already become a well-established image processing method [5]. Seminal works to demonstrate and formulate this approach at the early 90s [19, 0] laid out the foundation to many applications such as denoising, gap completion and computer aided quality control. Still, when comparing the PDE-based methods to non-pde-based filters see [1,,,3]) it is often noticed that by itself, each iteration step in the numerical solution of a PDE is intrinsically local when compared to a single iteration step in other filtering approaches. This limitation can be overcome by proposing a simple generalization to describe the nonlinear diffusion equation for extended neighborhoods. Furthermore, for unstructured grids, the digital TV filter [8] has been proposed by using graph connectivities to relate between neighboring pixels. We show a simple extension of nonlinear diffusion address: dbarash@research.haifa.ac.il D. Barash) /$ IMACS. Published by Elsevier B.V. All rights reserved. doi: /j.apnum

2 D. Barash / Applied Numerical Mathematics 5 005) 1 11 filtering to treat larger neighborhoods on structured grids, which can be further extended to unstructured grids [8]. The intermediate step that is outlined here, for generalizing the nonlinear diffusion equation to treat S + 1 S + 1)-neighborhood, is important both pedagogically and from an application perspective due to its simplicity in treating structured grids. In Section, the extension of adaptive smoothing and nonlinear diffusion from a 3 3 window to a5 5 window is demonstrated. This generalization is further carried out to treat S + 1 S + 1)- neighborhood in Section 3. Section 4 shows the merits of this generalization on example images, followed by the Conclusions.. Importance of extended neighborhood The extension of gradient based, edge-preserving smoothing to include information from non-nearest neighboring pixels is natural and has been considered before in various contexts and other approaches than the one proposed in this paper e.g., [5,8,4,]). Here, unlike these other approaches, we start from Saint-Marc Chen Medioni s adaptive smoothing [1] that was reformulated in [] for consistency with the diffusion equation, and extend the approach from the original 3 3 window to a window of arbitrary size. The adaptive smoothing approach is fundamental and intuitive. Given an image I t) x), where x = x 1,x ) denotes space coordinates, an iteration of adaptive smoothing yields: I t+1) x) = i= 1 j= 1 I t) x 1 + i, x + j)w t) +1 i= 1 +1 j= 1 wt), 1) where the convolution mask w t) is defined as: d t) w t) x 1,x ) ) x 1,x ) = exp, ) k where k is the variance of the Gaussian mask. In [1], d t) x 1,x ) is chosen to depend on the magnitude of the gradient computed in a 3 3 window: d t) x 1,x ) = G x 1 + G x, 3) where G x1,g x ) = I t) ) x 1,x ), It) x 1,x ), 4) x 1 x noting the similarity see also [4,14,] for further analogies) of the convolution mask w with the diffusion coefficient in anisotropic diffusion [19,5], or more specifically, the total variation in Rudin Osher Fatemi s original work [0] that demonstrated how edge-preserving smoothing can be achieved from energy minimization.

3 D. Barash / Applied Numerical Mathematics 5 005) Smoothing on 1D 3-neighborhood It was suggested in [1] that Eq. 1) is an implementation of anisotropic diffusion. Briefly sketched, let us consider the case of a one-dimensional signal I t x) and reformulate the averaging process as follows: with I t+1 x) = c 1 I t x 1) + c I t x) + c 3 I t x + 1), 5) c 1 + c + c 3 = 1. Therefore, it is possible to write the above iteration scheme as: I t+1 x) I t x) = c 1 I t x 1) I t x) ) + c 3 I t x + 1) I t x) ). 7) Taking c 1 = c 3 = c, this reduces to: I t+1 x) I t x) = c I t x 1) I t x) + I t x + 1) ), 8) which is a discrete approximation of the linear diffusion equation: I t = c I... Adaptive smoothing on 1D 3-neighborhood When the weights are space-dependent, one should write the weighted averaging scheme see [] for more details) as follows: I t+1 x) = ct x 1)I t x 1) + c t x)i t x 1) + c t x)i t x) + ct x + 1)I t x + 1) + c t x)i t x + 1), 10) with c t x 1) + c t x) + c t x) + ct x + 1) + c t x) = 1, 11) noting that the coefficients c 1 and c 3 of 5) are space-dependent, c 1 = cx h/) and c 3 = cx + h/) where h = 1, approximated by simple averages. Plugging 11) into 10) and rearranging as in []) leads to a discrete approximation to the nonlinear diffusion equation: I t = cx) I ). 1).3. Smoothing on 1D 5-neighborhood Back to non-adaptive smoothing, the averaging process can be extended to include second-neighbors: I t+1 x) = c 1 I t x ) + c I t x 1) + c 3 I t x) + c 4 I t x + 1) + c 5 I t x + ), 13) 6) 9)

4 4 D. Barash / Applied Numerical Mathematics 5 005) 1 11 with c 1 + c + c 3 + c 4 + c 5 = 1. Taking c 1 = c 5 = w, c = c 4 = w 1,andc 3 = 1 w w 1 this reduces to: I t+1 x) = w I t x ) + I t x + ) ) + 1 w w 1 )I t x) + w 1 I t x 1) + I t x + 1) ), 15) rearrangement of terms leads to: I t+1 x) I t x) = w I t x ) I t x) + I t x + ) ) 14) + w 1 I t x 1) I t x) + I t x + 1) ) 16) which is a discrete approximation of the linear diffusion equation: I t = w 1 1 I + w I, 17) where 1 denotes over a grid containing only the nearest-neighbors, and denotes over a grid containing only the second-neighbors. Typically w 1 >w since nearest-neighbors have more influence than second-neighbors..4. Adaptive smoothing on 1d 5-neighborhood Adaptive smoothing can be extended to treat 5-neighborhood in 1D as follows: c I t+1 t x ) + c t ) x) c x) = I t t x 1) + c t ) x) x ) + I t x 1) c + c t x)i t t x + 1) + c t ) x) c x) + I t t x + ) + c t ) x) x + 1) + I t x + ), 18) with c t x ) + c t x) + ct x 1) + c t x) + c t x) + ct x + 1) + c t x) + ct x + ) + c t x) = 1. 19) Plugging 19) into 18) and rearranging finally leads to: I t+1 x) I t x) = ct x + ) + c t x)[ I t x + ) I t x) ] ct x ) + c t x)[ I t x) I t x ) ] + ct x + 1) + c t x)[ I t x + 1) I t x) ] ct x 1) + c t x)[ I t x) I t x 1) ], 0) which is a consistent implementation of the nonlinear diffusion equation: I t = 1 w1 x 1,x ) 1 I ) + w x 1,x ) I ), 1)

5 D. Barash / Applied Numerical Mathematics 5 005) where we have used w instead of c, since the variable w was adopted in 17) instead of c in 9). Thus, the weights w 1 x 1,x ), w x 1,x ) are the nonlinear diffusion coefficients in the nearest-neighbors grid or second-neighbor grid, respectively, typically taken as: w 1, x 1,x ) = g 1, Ix 1,x ) ), ) where 1, I is the gradient magnitude on either the nearest-neighbors grid or the second-neighbors grid, respectively, and g 1, I ) is an edge-stopping function. This function is chosen to satisfy gx) 0whenx so that the diffusion is stopped across edges. Thus, a fundamental link between the nonlinear diffusion equation and edge-preserving smoothing filters is noticed in addition to the interesting relationships between nonlinear PDEs and morphological filters that have been explored in [6,18,15], for example). This link will be extended in the next section to a general S + 1 S + 1)- neighborhood. 3. Generalized adaptive smoothing and nonlinear diffusion on extended neighborhood Adaptive smoothing was introduced in [1] as a local process applying a 3 3 window at each iteration, as defined in 1). However, it is natural to extend this definition to an arbitrary, S + 1 S + 1) window I t+1) x) = +S +S i= S j= S I t) x 1 + i, x + j)w t) +S i= S +S j= S wt), 3) where i is the index corresponding to the x-direction, j is the index corresponding to the y-direction, and there is no coupling between them. Stability and the extremum principle hold since ) is a low pass filter, regardless of its nonlinear dependence on the data itself, with positive coefficients that are normalized. A discussion of adaptive smoothing and related filtering approaches without the restriction to nearest-neighbors in the weights is also available in [,8]. In the rest of this Section, the extended nonlinear diffusion is derived using the generalization of adaptive smoothing outlined in ). First, it is instructive to derive the extended nonlinear diffusion in one-dimension. Considering adaptive smoothing on 1D S + 1)-neighborhood, it is possible to generalize 18) by using vector notation to: c ) + cx)ˆ1 c+) + cx)ˆ1 I t+1 x) = ) I t ) + cx)i t x) + ) I t +), 4) where ˆ1 =[1, 1,...,1] is the unity vector, denotes the dot product, c ) =[cx S),cx S + 1),...,cx 1)], c +) =[cx + 1), cx + ),..., cx + S)], I ) =[Ix S),Ix S + 1),..., I x 1)], I +) =[Ix+ 1), I x + ),..., I x + S)], cx) and Ix) are scalars, Ix) being the gray-level intensity at the point of interest x. By analogy to 19), or adaptive smoothing in 1D 3-neighborhood outlined in [], normalization of the weights can be written in vector notation as: ) c ) + cx)ˆ1 c+) + cx)ˆ1 ˆ1 + cx) + and by analogy to 0), or 13) of [], we obtain: I t+1 x) I t c+) + cx)ˆ1 x) = ) ˆ1 = 1, 5) ) [ I t +) I t x)ˆ1 ] c ) + cx)ˆ1 ) [I t x)ˆ1 I t )], 6)

6 6 D. Barash / Applied Numerical Mathematics 5 005) 1 11 Fig. 1. Two instances for calculating gradients and taking weights with respect to point P, in the case S = 3. Marked points correspond to i = 1, j = 3 of 3), 9) left figure), and i =, j = of 3), 9) right figure). which is an implementation of the nonlinear diffusion equation: Ix) = wx) Ix) ), 7) t where =[ S, S+1,..., S ] is a vector containing gradients taken at different neighboring configurations i.e., nearest-neighbors, second-neighbors, etc.) and w =[w S,w S+1,...,w S ] are the nonlinear diffusion coefficients. It is also possible to write 7) as: Ix) S = j wj x) j Ix) ), 8) t j= S expanding the vector notation used in 7). Second, the generalization of adaptive smoothing in two-dimensions written in 3) leads to the extended nonlinear diffusion in two-dimensions by simple analogy to 7). Taking matrices instead of vectors for,w leads to the extended nonlinear diffusion: Ix 1,x ) = wx 1,x ) Ix 1,x ) ), 9) t where denotes the scalar product between two matrices, and, w are S + 1) S + 1) matrices that correspond to different neighbor combinations with respect to the center pixel of interest see Fig. 1 for two illustrative examples). The generalized adaptive smoothing 3) is a discrete approximation of the extended nonlinear diffusion 9). Conceptually, further points can be used to obtain higher order accurate schemes for the discretization, although it is not clear how important is the role of accuracy in common image processing tasks. It is noted that in practice, S need not be taken too large i.e., S 3), otherwise the generalized adaptive smoothing becomes an inaccurate representation of the extended diffusion equation given in 9). 4. Comparison of denoising with extended vs. standard nonlinear diffusion equation Following the formulation of the extended nonlinear diffusion in 9), we would like to examine the merits of this generalization with respect to the standard nonlinear diffusion equation. Other numerical

7 D. Barash / Applied Numerical Mathematics 5 005) Fig.. Two possible stencils for the nearest-neighbors case S = 1): Simple 5-node stencil left figure), and full 9-node stencil right figure). The stencils are similar to the ones in [8, Fig. 1]. geometry extensions [3] and nonlinear filtering approaches that are connected to nonlinear diffusion but are not a direct realization of a PDE e.g., [4,13,1,9 11,1 3]), are discussed elsewhere. Here, a comparison is performed with the purpose of investigating the advantages of using an extended neighborhood over nearest-neighbors in the finite-difference numerical solution of the nonlinear diffusion equation. As an illustrative example, the Additive Operator Splitting AOS) numerical scheme along with a pre-smoothing to ensure the well-posedness of the PDE approach [7]), as described in [6], will be used for the comparison. It should be noted, however, that the formulation outlined in this paper is applicable to any numerical scheme that solves the continuous equation 9). Moreover, it should be stressed that the AOS uses the simple 5-node stencil as in Fig. left), without taking into account diagonal neighboring pixels when calculating gradients, whereas the generalized adaptive smoothing 3) and extended nonlinear diffusion 9) support a full 9-node stencil. The full stencil is shown in Fig. right) and may possess some advantages also drawn in [8, Fig. 1], although actual experiments in [8] were performed with the 5-node stencil as here, see [8, remark to caption of Fig. 1]). Nevertheless, for simplicity and given the fact that the AOS scheme is an efficient implementation of nonlinear diffusion filtering, we will perform the comparison with the 5-point stencil in Fig. left) extending it to treat second-neighbors S = ). Thus, the comparison here is performed using the AOS scheme between the cases of nearest-neighbors S = 1) and second-neighbors S = ), in denoising synthetic images. In our experiments the diagonal elements in the window will not influence the filtering because of the splitting imposed by the AOS scheme. The AOS was devised in [6] for an efficient implementation of nonlinear diffusion image filtering, and has been suggested in the past for solving the Navier Stokes equations in [16,17]. In Figs. 3 and 4 two noisy synthetic images are assessed for the comparison. Fig. 5 presents the same example on a natural image. The number of arithmetic operations when using AOS with S = 1, Figs. 3B), 4B) and 5B), along with 40 iterations is similar to the number of arithmetic operations when using AOS with S =, Figs. 3C), 4C) and 5C), along with 0 iterations. Intentionally, a large time step of t = 4.0 which is amenable with the AOS scheme from the point of view of stability and accuracy [6]) was chosen for the comparison in order to accentuate the differences between S = 1 and S =. In these examples Figs. 3 5) we observe the merits of using an extended neighborhood: for roughly the same complexity, better denoising was achieved, albeit a tradeoff in sharpness. Thus, in some image processing applications, there may be an advantage for solving the nonlinear diffusion equation with an extended neighborhood.

8 8 D. Barash / Applied Numerical Mathematics 5 005) 1 11 A) B) C) Fig. 3. Comparison Example I Pants): the additive operator splitting AOS) [6] with a fixed pre-smoothing of σ = 0.1 is applied to denoise the original image in A), with a time step of t = 4.0. Nearest-neighbors S = 1) window was used for 40 iterations in B), whereas second-neighbors S = ) window was used for 0 iterations in C). 5. Conclusions The formulation outlined in this paper is directly applicable to images on structured grids. It generalizes the nonlinear diffusion equation, based on the adaptive smoothing approach, to treat extended

9 D. Barash / Applied Numerical Mathematics 5 005) A) B) C) Fig. 4. Comparison Example II Shapes): the additive operator splitting AOS) [6] with a fixed pre-smoothing of σ = 0.1 is applied to denoise the original image in A), with a time step of t = 4.0. Nearest-neighbors S = 1) window was used for 40 iterations in B), whereas second-neighbors S = ) window was used for 0 iterations in C). neighborhoods that are larger than a 3 3 window. This formulation can be used in two distinct directions: the first is to construct a robust iterative procedure that performs broad kernel filtering based on 3) with a dynamic window [3], and the second is to extend the PDE solution approach for the diffusion process. Here, we have experimented with the latter direction, extending the PDE solution approach. The next level of complexity for generalizing the diffusion process to unstructured grids, based on graph connectivities, has already been devised in [8] for the case of total variation diffusivities and merits further investigation.

10 10 D. Barash / Applied Numerical Mathematics 5 005) 1 11 A) B) C) Fig. 5. Comparison Example III Grasshopper): the additive operator splitting AOS) [6] with a fixed pre-smoothing of σ = 0.1 is applied to denoise the original image in A), with a time step of t = 4.0. Nearest-neighbors S = 1) window was used for 40 iterations in B), whereas second-neighbors S = ) window was used for 0 iterations in C). Acknowledgement This work was conducted at the Genome Diversity Center and supported by the Institute of Evolution, University of Haifa, Israel.

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